$$\frac { x ( y + z ) + z ( y - x ) } { x ^ { 2 } + x y + x z + y z }$$ Which of the following is the simplified form of this expression? A) $\frac { x } { x + y }$ B) $\frac { y } { x + y }$ C) $\frac { z } { x + z }$ D) $\frac { y } { x + z }$ E) $\frac { y } { y + z }$
For positive real numbers x and y, $$\begin{aligned}
x \cdot y & = 5 \\
x ^ { 2 } + y ^ { 2 } & = 15
\end{aligned}$$ Given this, what is the value of the expression $x ^ { 3 } + y ^ { 3 }$? A) 40 B) 45 C) 50 D) 60 E) 75
Let x and y be real numbers such that $$\begin{aligned}
& x ^ { 2 } - 4 y = - 7 \\
& y ^ { 2 } - 2 x = 2
\end{aligned}$$ Given this, what is the sum $x + y$? A) 3 B) 4 C) 5 D) $\frac { 4 } { 3 }$ E) $\frac { 5 } { 3 }$
Let x be a real number such that $$( \sqrt { 7 } + \sqrt { 3 } ) ^ { x } = 4$$ Given this, which of the following is the expression $( \sqrt { 7 } - \sqrt { 3 } ) ^ { x }$ equal to? A) $2 ^ { - x }$ B) $2 ^ { - x + 1 }$ C) $4 ^ { x }$ D) $4 ^ { x - 1 }$ E) $4 ^ { x + 1 }$
$$\left. \begin{array} { l }
2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\
2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 )
\end{array} \right\}$$ For positive integers a and b that satisfy both congruences simultaneously, what is the minimum value of the sum $a + b$? A) 3 B) 4 C) 5 D) 6 E) 7
Let x and y be real numbers with $-1 < y < 0 < x$. Which of the following statements are always true? I. $x + y > 0$ II. $x - y > 1$ III. $x \cdot ( y + 1 ) > 0$ A) Only I B) Only III C) I and II D) I and III E) II and III
The operation $\Delta$ is defined on the set of real numbers for all real numbers a and b as $$a \Delta b = a ^ { 2 } + 2 ^ { b }$$ Given that $2 \Delta ( 1 \Delta x ) = 12$, what is x? A) $\frac { 1 } { 2 }$ B) $\frac { 2 } { 3 }$ C) $\frac { 1 } { 4 }$ D) 1 E) 2
Let $Z$ be the set of integers. The function $f : Z \rightarrow Z$ is defined as $$f ( x ) = \begin{cases} x - 1 , & \text{if } x < 0 \\ x + 1 , & \text{if } x \geq 0 \end{cases}$$ Accordingly, I. f is one-to-one. II. f is onto. III. The range of f is $Z \backslash \{ 0 \}$. Which of these statements are true? A) Only I B) Only II C) Only III D) I and II E) I and III
$$\begin{aligned}
& f ( x ) = | 2 x - 5 | \\
& g ( x ) = | x + 1 |
\end{aligned}$$ The functions are given. Accordingly, what is the sum of the x values that satisfy the equation $( g \circ f ) ( x ) = 3$? A) $-3$ B) $-1$ C) 0 D) 2 E) 5
A function f defined on the set of real numbers satisfies the inequality $$f ( x ) < f ( x + 2 )$$ for every real number x. Accordingly, I. $f ( 1 ) < f ( 5 )$ II. $| f ( - 1 ) | < | f ( 1 ) |$ III. $f ( 0 ) + f ( 2 ) < 2 \cdot f ( 4 )$ Which of these statements are always true? A) Only I B) Only II C) I and III D) II and III E) I, II and III
Let a and b be positive integers. The sum of the coefficients of the polynomial $$P ( x ) = ( x + a ) \cdot ( x + b )$$ is 15. What is the sum $a + b$? A) 10 B) 9 C) 8 D) 7 E) 6
$$\begin{aligned}
& P ( x ) = x ^ { 2 } - 2 x + m \\
& Q ( x ) = x ^ { 2 } + 3 x + n
\end{aligned}$$ polynomials are given. These two polynomials have a common root and the roots of the polynomial $P(x)$ are equal, so what is the sum $m + n$? A) $-5$ B) $-3$ C) 2 D) 4 E) 5
$$y = x ^ { 2 } - 2 ( a + 1 ) x + a ^ { 2 } - 1$$ If the parabola is tangent to the line $y = 1$, what is a? A) $\frac { -3 } { 2 }$ B) $\frac { -3 } { 4 }$ C) 0 D) 1 E) 2
A florist has roses of 5 different colors in large quantities and 2 types of vases. A customer wants to buy a total of 3 roses of 2 different colors and 1 vase. In how many different ways can this customer make the purchase? A) 15 B) 20 C) 25 D) 40 E) 50
A bag contains 5 red and 4 white marbles. When 3 marbles are drawn randomly from this bag at the same time, what is the probability that there are at most 2 marbles of each color? A) $\frac { 2 } { 3 }$ B) $\frac { 3 } { 4 }$ C) $\frac { 5 } { 6 }$ D) $\frac { 7 } { 8 }$ E) $\frac { 8 } { 9 }$
ABCD is a square, $|BE| = 5$ cm, $|EC| = 7$ cm, $m(\widehat{EAC}) = x$. According to the given information, what is $\tan x$? A) $\frac { 4 } { 13 }$ B) $\frac { 6 } { 13 }$ C) $\frac { 9 } { 13 }$ D) $\frac { 5 } { 17 }$ E) $\frac { 7 } { 17 }$
On the set of complex numbers $$f ( z ) = 1 - 2 z ^ { 6 }$$ a function is defined. For $z _ { 0 } = \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right)$, what is $f \left( z _ { 0 } \right)$? A) $1 + i$ B) $2i$ C) $1 - i$ D) $-1$ E) $3$
$$( | z | + z ) \cdot ( | z | - \bar { z } ) = i$$ Which of the following is the imaginary part of the complex number z that satisfies the equation? A) $\frac { 2 } { | z | }$ B) $\frac { 1 } { | z | }$ C) $\frac { - | z | } { 2 }$ D) $\frac { 1 } { 2 | z | }$ E) $- | z |$
For the complex number $z = a + ib$ whose distance to the number 1 is 2 units and whose distance to the number i is 3 units, what is the difference $a - b$? A) $\frac { 3 } { 2 }$ B) $\frac { 5 } { 2 }$ C) $\frac { 7 } { 2 }$ D) $\frac { 4 } { 3 }$ E) $\frac { 7 } { 3 }$
$$\sum _ { n = 4 } ^ { 9 } \left( \prod _ { k = 1 } ^ { n } \frac { k + 1 } { k } \right)$$ What is the result of this operation? A) 45 B) 48 C) 50 D) 52 E) 54
The sequence $\left( a _ { n } \right)$ $$a _ { n } = \begin{cases} 2 ^ { n } + 1 , & n \equiv 0 ( \bmod 2 ) \\ 2 ^ { n } - 1 , & n \equiv 1 ( \bmod 2 ) \end{cases}$$ is defined in the form. Accordingly, what is the value of the expression $\frac { a _ { 9 } - a _ { 7 } } { a _ { 8 } - 4 \cdot a _ { 6 } }$? A) $-2 ^ { 8 }$ B) $-2 ^ { 7 }$ C) $-2 ^ { 6 }$ D) $-2 ^ { 5 }$ E) $-2 ^ { 4 }$
Below, a sequence of circles drawn side by side is given. In this sequence; the radius of the first circle is 4 units and the radius of each subsequent circle is half the radius of the previous circle. What is the sum of the circumferences of all circles in this sequence in units? A) $15 \pi$ B) $16 \pi$ C) $18 \pi$ D) $\frac { 31 \pi } { 2 }$ E) $\frac { 33 \pi } { 2 }$
Let a, b and c be positive real numbers, $$\left[ \begin{array} { l l }
a & b \\
0 & c
\end{array} \right] \cdot \left[ \begin{array} { l l }
a & b \\
0 & c
\end{array} \right] = \left[ \begin{array} { l l }
1 & 2 \\
0 & 4
\end{array} \right]$$ The matrix equation is given. Accordingly, what is the sum $a + b + c$? A) $\frac { 11 } { 3 }$ B) $\frac { 7 } { 4 }$ C) 4 D) 5 E) 6
For a matrix A with multiplicative inverse $A^{-1}$, $$\left[ \begin{array} { l l } 2 & 1 \end{array} \right] \cdot \left[ \begin{array} { l l } 1 & 0 \\ 3 & 1 \end{array} \right] ^ { - 1 } \cdot \left[ \begin{array} { l } 1 \\ 4 \end{array} \right] = [ a ]$$ In the matrix equation, what is a? A) 1 B) 2 C) 3 D) 4 E) 5
$$\begin{aligned}
& A = \left[ \begin{array} { l l }
2 & 3 \\
1 & 2
\end{array} \right] \\
& B = \left[ \begin{array} { l l }
1 & 2 \\
0 & 5
\end{array} \right]
\end{aligned}$$ With the matrix notation $$( 2 A - B ) \cdot \left[ \begin{array} { l }
x \\
y
\end{array} \right] = \left[ \begin{array} { l }
1 \\
0
\end{array} \right]$$ Which of the following is the system of linear equations? A) $\begin{aligned} & x - 4 y = 0 \\ & 2 x - y = 1 \end{aligned}$ B) $\begin{aligned} & x + 2 y = 0 \\ & 2 x - 3 y = 1 \end{aligned}$ C) $\begin{aligned} & 2 x + y = 1 \\ & x - y = 0 \end{aligned}$ D) $\begin{aligned} & 3 x - 2 y = 1 \\ & 2 x + y = 0 \end{aligned}$ E) $\begin{aligned} & 3 x + 4 y = 1 \\ & 2 x - y = 0 \end{aligned}$
$$\lim _ { x \rightarrow 1 ^ { + } } ( x - 1 ) \cdot \ln \left( x ^ { 2 } - 1 \right)$$ What is the value of this limit? A) $\frac { -1 } { 2 }$ B) $-2$ C) 0 D) 1 E) 4
For a function f defined on the set of real numbers $$\begin{aligned}
& \lim _ { x \rightarrow 3 ^ { + } } f ( x ) = 1 \\
& \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = 2
\end{aligned}$$ Given this, what is the value of the limit $\lim _ { x \rightarrow 2 ^ { + } } \frac { f ( 2 x - 1 ) + f ( 5 - x ) } { f \left( x ^ { 2 } - 1 \right) }$? A) $\frac { -1 } { 2 }$ B) $\frac { 3 } { 2 }$ C) 1 D) 3 E) 4
$$f ( x ) = \begin{cases} 1 , & x \leq 1 \\ x ^ { 2 } + a x + b , & 1 < x < 3 \\ 5 , & x \geq 3 \end{cases}$$ If the function is continuous on the set of real numbers, what is the difference $a - b$? A) $-4$ B) $-1$ C) 2 D) 3 E) 5
For functions f and g defined on the set of real numbers $$\begin{aligned}
& f ( g ( x ) ) = x ^ { 2 } + 4 x - 1 \\
& g ( x ) = x + a \\
& f ^ { \prime } ( 0 ) = 1
\end{aligned}$$ Given this, what is a? A) $-2$ B) $\frac { -1 } { 4 }$ C) 1 D) $\frac { 3 } { 2 }$ E) 3
$$f ( 2 x + 5 ) = \tan \left( \frac { \pi } { 2 } x \right)$$ For the function $f$ given by the equality, what is the value $f ^ { -1 } ( 1 )$? A) $\frac { \pi } { 2 }$ B) $\frac { \pi } { 4 }$ C) $\pi$ D) $2 \pi$ E) $3 \pi$
A third-degree real-coefficient polynomial function $P(x)$ with leading coefficient 1 has two of its roots as $-5$ and $2$. If $P(x)$ has a local extremum at the point $x = 0$, what is the third root? A) $\frac { 1 } { 2 }$ B) $\frac { 3 } { 2 }$ C) $\frac { 7 } { 3 }$ D) $\frac { -5 } { 2 }$ E) $\frac { -10 } { 3 }$
Below, the graph of the derivative of a function f that is defined and continuous on the set of real numbers is given. Accordingly, I. $f ( 2 ) - f ( 1 ) = -2$. II. The function f has a local maximum at the point $x = 0$. III. The second derivative function is defined at the point $x = 0$. Which of the following statements are true? A) Only I B) Only III C) I and II D) II and III E) I, II and III
For $x > 0$; if the point $(a, b)$ on the graph of the curve $y = 6 - x^2$ is closest to the point $(0, 1)$, what is b? A) $\frac { 3 } { 2 }$ B) $\frac { 5 } { 2 }$ C) $\frac { 7 } { 2 }$ D) $\frac { 9 } { 2 }$ E) $\frac { 11 } { 2 }$